Simple vs. Optimal Congestion Pricing

Abstract

Congestion pricing has emerged as an effective tool for mitigating traffic congestion, yet implementing welfare or revenue-optimal dynamic tolls is often impractical. Most real-world congestion pricing deployments, including New York City's recent program, rely on significantly simpler, often static, tolls. This discrepancy motivates the question of how much revenue and welfare loss there is when real-world traffic systems use static rather than optimal dynamic pricing. We address this question by analyzing the performance gap between static (simple) and dynamic (optimal) congestion pricing schemes in two canonical frameworks: Vickrey's bottleneck model with a public transit outside option and its city-scale extension based on the Macroscopic Fundamental Diagram (MFD). In both models, we first characterize the revenue-optimal static and dynamic tolling policies, which have received limited attention in prior work. In the worst-case, revenue-optimal static tolls achieve at least half of the dynamic optimal revenue and at most twice the minimum achievable system cost across a wide range of practically relevant parameter regimes, with stronger and more general guarantees in the bottleneck model than in the MFD model. We further corroborate our theoretical guarantees with numerical results based on real-world datasets from the San Francisco Bay Area and New York City, which demonstrate that static tolls achieve roughly 80-90% of the dynamic optimal revenue while incurring at most a 8-20% higher total system cost than the minimum achievable system cost.

Figures (11)

• Figure 1: Equilibrium waiting time profiles in the bottleneck model without an outside option (left) and with a public transit outside option in a mixed-mode equilibrium where both car and transit are used (right) in the setting when $\mu < \lambda$. The slopes $e$ and $L$ denote the normalized schedule-delay penalties for early and late arrivals, respectively, and $[t_1,t_2]$ represents users’ desired bottleneck crossing times.
• Figure 2: Depiction of the revenue optimal dynamic tolling policy. Here, $[t_A^{*}, t_D^{*}]$ denotes the equilibrium interval over which users pass the bottleneck under the dynamic revenue-optimal tolling policy $\tau_d^*(\cdot)$, analogous to the no-toll equilibrium interval $[t_A, t_D]$ shown in the right of Figure \ref{['fig:equilibrium_both_models']}. The sub-interval $[t_B^{*}, t_C^{*}]$ corresponds to the middle of the rush, during which the toll is constant and equal to $z_T - z_C$.
• Figure 3: Sum of the waiting plus toll costs at equilibrium. The time points $t_A', t_B', t_C', t_D'$ are determined endogenously by the tolling policy and may, in general, differ from the corresponding time points shown in Figure \ref{['fig:equilibrium_both_models']} (right) and Figure \ref{['fig:rev-opt-dynamic-tolls']}, which correspond to the no-toll and revenue-optimal dynamic toll settings, respectively.
• Figure 4: Depiction of the triangular MFD relating the system outflow capacity to total vehicle accumulation. The slope of the segment connecting $(0,0)$ to any point on the curve determines the average system speed, equal to average trip distance $D$ times the slope. The MFD consists of an uncongested regime in which the system operates at free-flow with a speed $v_f$ up to the critical threshold $n_c$. Beyond $n_c$, the system enters a congested regime in which speeds decline, eventually dropping to zero at the jam accumulation level $n_j$.
• Figure 5: Depiction of the revenue (left) and system cost (right) ratios of a set of static and dynamic tolling policies, normalized by the corresponding dynamic revenue-optimal and system-cost-optimal benchmarks, as the discomfort multiplier $\eta$ is varied for the Bay Bridge case study.

Theorems & Definitions (11)

• Proposition 1: Two-Mode Equilibrium GONZALES20121519
• Proposition 2: Revenue-Optimal Static Tolls
• proof
• Theorem 1: Revenue-Optimal Dynamic Tolls
• proof
• Theorem 2: Revenue Ratio: Static vs. Dynamic Revenue-Optimal Tolls
• Theorem 3: System Cost Comparison
• Proposition 3: Unbounded System Cost Ratios
• Theorem 4: Revenue-Optimal Dynamic Tolls under MFD
• Theorem 5: Revenue under Static Tolls in MFD Framework